Recurring payments not being recorded in CiviCRM

After a recent update -- we're now on the latest version (6.7.9) -- we're finding that recurring payments are not being saved correctly in CiviCRM. Transcribing from what I've written on Mattermost:

The payments go through in Stripe, and from the Stripe side the webhook is marked as successful. But the data doesn't make it into the contact records in CiviCRM and the civicrm/a#/paymentprocessorWebhook page indicates those webhooks fail:

• Invalid input data (not an object) #0 /app/files/civicrm/ext/com.drastikbydesign.stripe/CRM/Core/Payment/StripeIPN.php(176): CRM_Core_Payment_StripeIPN->exception('Invalid input d...')

NOTE: I have found that going into Stripe and resending the webhook (even though it says it was successful) will then record the contribution in CiviCRM.

One more data point: for one of these recurring payments, a user reported seeing it in CiviCRM but that it mysteriously disappeared. I'm not sure what could cause that...